I know that there are plants that can sprout from cuttings, so I wonder about this. I have a mulching mower that deposits clippings back on the lawn. There are some weeds like thistles and violets that seem to thrive the more I mow them. Is it because my mower is propagating them?
Highly unlikely that your mulching mower is causing cuttings to grow. The cut weed parts won't have good soil contact and will dry out and die. Weeds that have gone to seed stand a better chance of seeding, but they'll do that regardless of your mower.
Cutting too short, on the other hand, can definitely favor weeds over grass, where grass can shade out weeds if permitted to grow tall enough. Fairly stock recommendation is 3"/75mm cut height, and cut when it gets to 4"/100mm. Apart from tipping the balance in favor of grass, that also permits enough root growth for the lawn to be more drought-tolerant than very short grass.